12th Swedish Mathematical Society Problems 1972

1.  Find the largest real number a such that x - 4y = 1, ax + 3y = 1 has an integer solution.

2.  A rectangular grid of streets has m north-south streets and n east-west streets. For which m, n > 1 is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?

3.  A steak temperature 5^o is put into an oven. After 15 minutes, it has temperature 45^o. After another 15 minutes it has temperature 77^o. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.

4.  Put x = log₁₀2, y = log₁₀3. Then 15 < 16 implies 1 - x + y < 4x, so 1 + y < 5x. Derive similar inequalities from 80 < 81 and 243 < 250. Hence show that 0.47 < log₁₀3 < 0.482.

5.  Show that ∫ ₀¹ (1/(1 + xⁿ)) dx > 1 - 1/n for all positive integers n.

6.  a₁, a₂, a₃, ... and b₁, b₂, b₃, ... are sequences of positive integers. Show that we can find m < n such that a_m ≤ a_n and b_m ≤ b_n. 

Solutions

Problem 1

Find the largest real number a such that x - 4y = 1, ax + 3y = 1 has an integer solution.

Answer

1

Solution

For a = 1, we have x = 1, y = 1. Suppose a > 1. Then a(4y+1) = 1-3y, so y = -(a-1)/(4a+3) < 0. But a-1 < 4a+3 (since a is positive), so y > -1. Contradiction. So there are no solutions with a > 1.

Thanks to Suat Namli

Problem 2

A rectangular grid of streets has m north-south streets and n east-west streets. For which m, n > 1 is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?

Answer

mn even

Solution

Suppose a path is possible. Let N,S,E,W be the number of moves north, south, east, west. Then the total number of moves N + S + E + W = mn. Since the path is closed we have E = W and N = S. Hence N + S + E + W is even and hence mn is even.

wlog suppose n is even. The diagram shows one way of making the drive using a comb shape.

Problem 3

A steak temperature 5^o is put into an oven. After 15 minutes, it has temperature 45^o. After another 15 minutes it has temperature 77^o. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.

Answer

205^o

Solution

The steak temperature is obviously A - Be^-kt, where A is the oven temperature, B and k are constants and t is the time. (One can write down the differential equation and solve it). Put h = e^-15k.

So we have A - B = 5, A - Bh = 45, A - Bh2 = 77. Subtracting, B(1 - h) = 40, Bh(1 - h) = 32. Hence h = 4/5, B = 200, A = 205.

Problem 5

Show that ∫ ₀¹ (1/(1 + xⁿ)) dx > 1 - 1/n for all positive integers n.

Solution

We have 1/(1+xⁿ) = 1 - xⁿ/(1+xⁿ) > 1 - xⁿ (for x in (0,1)). Hence ∫ ₀¹ (1/(1 + xⁿ)) dx > 1 - ∫ ₀¹ xⁿ dx = 1 - 1/(n+1) > 1 - 1/n.

Thanks to Thomas Linhart